Why is my REST method not being called by this jQuery?

I have a similar question about jQuery button click handler code not being fired at all here.

In this case, it is being fired (when the jQuery is added to a static page (Index.cshtml)), but my REST method is still not being reached. The console message explaining why is:

The resource cannot be found.

Description: HTTP 404. The resource you are looking for (or one of its dependencies) could have been removed, had its name changed, or is temporarily unavailable. Please review the following URL and make sure that it is spelled correctly.

UPDATE 5

Note: The maximum bounty (200 points, I think it is) will be awarded to whoever can solve this dilemma. If more than one person does, then the bounty goes to whoever solves it best (which basically to me means in the most straightforward and easy-to-implement manner.

In the interest of full disclosure and desperation, here is the entire contents (with boring/meaningless parts elided with ellipsis dots) of the first page that displays when the WEB API app runs (from \Views\Home\Index.cshtml: